JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, C02021, doi:10.1029/2002JC001707, 2004
Examination of reference concentration under waves
and currents on the inner shelf
Guan-hong Lee1 and W. Brian Dade2
Institute of Theoretical Geophysics, Cambridge University, Cambridge, UK
Carl T. Friedrichs
Virginia Institute of Marine Science, College of William and Mary, Gloucester Point, Virginia, USA
Chris E. Vincent
School of Environmental Sciences, University of East Anglia, Norwich, UK
Received 8 November 2002; revised 31 March 2003; accepted 12 November 2003; published 24 February 2004.
[1] We examine reference concentration using three different data sets of near-bed
suspended sediment concentration observed under combined waves and currents. The data
include observations made at 15 and 20 m depth off Dounreay, Scotland, UK, and
observations obtained at 13 m depth off Duck, North Carolina, USA. These data
accommodate different dynamic conditions (from wave-dominated conditions at
Dounreay to wind-driven, current-dominated conditions at Duck) and sediment properties
(median size of bed sediment ranging from 120 to 350 mm). Near-bed concentration
profiles to elevations of about 80 cm were obtained using acoustic backscatter sensors
with 1 cm resolution. The reference concentrations (Cr) at 1 cm were then evaluated by
regressing the observed suspended sediment concentrations against a Rouse-type model.
Bed shear stresses associated with each estimate of Cr were estimated using the wavecurrent interaction model of Grant and Madsen. Existing equations for reference
concentration based on shear stress alone fail to accommodate all Cr estimates from
different environments. We introduce a new empirical relationship between Cr and the
product of Shields and inverse Rouse numbers. These dimensionless parameters represent
the ratio of bed shear stress and submerged particle weight and the ratio of shear velocity
and particle settling velocity, respectively. The new formula adjusts the amount of mobile
sediment at the bed (related to the Shields number) to that available for suspension at
the reference height (related to the inverse Rouse number). The new formula for reference
concentration accommodates observations from different environments, suggesting that it
INDEX TERMS: 4558 Oceanography:
may have wide applicability on sandy inner shelves.
Physical: Sediment transport; 3022 Marine Geology and Geophysics: Marine sediments—processes and
transport; 4211 Oceanography: General: Benthic boundary layers; 4219 Oceanography: General: Continental
shelf processes; 4546 Oceanography: Physical: Nearshore processes; KEYWORDS: reference concentration,
suspension, sediment, Shields parameter, Rouse number
Citation: Lee, G., W. B. Dade, C. T. Friedrichs, and C. E. Vincent (2004), Examination of reference concentration under waves and
currents on the inner shelf, J. Geophys. Res., 109, C02021, doi:10.1029/2002JC001707.
1. Introduction
[2] On the inner shelf, sediment resuspension and transport typically occur as the result of the combined action of
waves and currents. Many models used in shelf sediment
transport applications predict the time-averaged profile of
sediment concentration for combined waves and currents by
1
Now at Korea Ocean Research and Development Institute, Ansan,
Korea.
2
Now at Department of Earth Sciences, Dartmouth College, Hanover,
New Hampshire, USA.
solving the steady state diffusion equation [e.g., Smith,
1977; Sleath, 1984; Glenn and Grant, 1987; Lee et al.,
2002]. In order to achieve these predictions, two components must be prescribed. One is the amount of sediment
available for suspension at a specified, near-bed elevation
(the reference concentration), and the other is the vertical
distribution of suspended sediment. In this paper, we are
concerned with reference concentration.
[3] In a seminal paper, Smith [1977] related the reference
concentration Cr to excess shear stress Se in the form
Cr ¼
Copyright 2004 by the American Geophysical Union.
0148-0227/04/2002JC001707$09.00
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Cb g o S e
;
ð1 þ go Se Þ
ð1Þ
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where Se = (tb tcr)/tcr, tb is the bed shear stress, tcr is
the critical shear stress for the bed material with
characteristic grain size ds, Cb is the relative concentration
of sediment in the bed (ffi0.65), and go is a dimensionless
resuspension coefficient. The seabed shear stress, tb, is
related to the shear velocity u* = (tb/r)1/2, where r is the
water density.
[4] Equation (1) is based on an expression first proposed
by Yalin [1963] and extended by Smith [1977] to accommodate the physical constraint that Cr can only approach
(and not exceed) Cb as excess shear stress Se becomes very
large. Under typical conditions, however, goSe 1, and
equation (1) is approximately given by
Cr ¼ Cb g o
!
u2*sf u2*cr
;
u2*cr
ð2Þ
where u*sf and u*cr are skin-friction and critical shear
velocity for initiation of sediment motion, respectively.
This approach has been widely used both under
unidirectional flow in river and estuarine environments
and under combined waves and currents in the coastal
environment [Madsen et al., 1993; Webb and Vincent,
1999; Green et al., 2000; Rose and Thorne, 2001].
However, values of the crucial resuspension coefficient go
reported by various workers vary from 105 to 102
[Drake and Cacchione, 1989; Hill et al., 1988; Webb and
Vincent, 1999] and, under some conditions, decrease
with increasing skin-friction shear velocity [Vincent and
Downing, 1994; Vincent and Osborne, 1995].
[5] Van Rijn [1984] introduced a similar empirical
expression between Cr and the excess shear stress, Se, in
the form
Cr ¼ 0:015
ds Se1:5
;
zr D0:3
*
ð3Þ
where zr is the height at which the reference concentration is
measured and D* is the particle parameter (= ds[(s 1)g/
n2]1/3, where g is the acceleration of gravity, n is kinematic
viscosity, and s is the density of sediment, rs, relative to sea
water, r). Rose and Thorne [2001] examined equation (3)
using sediment concentration data collected in an estuary
and reported that predictions by equation (3) were within a
factor of two of the measurements.
[6] Nielsen [1986] introduced an alternative relationship
between Cr and the dimensionless skin-friction Shields
parameter, qsf, based on both field and laboratory data
pertaining to relatively flat, mobile beds under waves. This
relationship is given simply as
Cr ¼ 0:005q3sf ;
ð4Þ
u2*sf
ðs 1Þgds
ð5Þ
where
qsf ¼
in terms of the time-averaged quantity u*sf2 and the
submerged grain weight [(rs r)gds]. Black and Rosenberg
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[1991] and Green and Black [1999] found equation (4) to
perform well under conditions of shoaling and broken
waves over flat beds. Equation (4) was also found to
perform well over rippled beds if the flow intensity
appearing in the numerator of equation (5) was corrected
to accommodate the effects of the ripples [Green and Black,
1999]. However, Webb and Vincent [1999] found no
dependence of Cr on the Shields parameter.
[7] The approaches represented by equations (1)– (5) are
based on the concept that relative fluid shear stress alone
determines the amount of near-bed sediment available for
suspension at a finite elevation zr above the bed. If zr is
chosen inappropriately, however, then the dynamics governing the vertical structure of the suspended sediment must be
considered [Van Rijn, 1984]. Even more fundamentally, we
suggest that not all mobile sediment is available for suspension. In this paper, we address these issues by introducing the inverse Rouse number which represents the ratio of
the skin friction velocity u*sf and the characteristic settling
velocity ws of bed material.
[8] In the next section, we develop these perspectives in
more detail. Three data sets of suspended sediment concentration encompassing different wave, current and grain size
regimes are then introduced (section 3) and considered
(sections 4 and 5) in the light of existing and new ideas
about physical constraints on a reference concentration
evaluated in consistent terms. In section 6, we conclude
with a brief discussion and review of key findings.
2. Analysis
[9] Sediment particles of a given size begin to move
when the characteristic shear velocity just exceeds the
critical shear velocity for initiation of motion. The threshold
condition for initiation of motion is given by the critical
Shields parameter, qcr, and is empirically determined as
[e.g., Miller et al., 1977; Dyer, 1986; Nielsen, 1992]
qcr ¼ u2*cr =½ðs 1Þgds :
ð6Þ
Mobile sediments enter into relatively continuous suspension, on the other hand, only if the shear velocity exceeds
the characteristic settling velocity of the particles under
consideration. Thus a suspension criterion can be expressed
in terms of the Shields parameter and defined as [Bagnold,
1966; Francis, 1973]
qs ¼ w2s =½ðs 1Þgds :
ð7Þ
[10] Figures 1a and 1b show the threshold conditions for
incipient motion and suspension in terms of the Shields
parameter, shear velocity and settling velocity [Allen, 1985;
Dyer, 1986; Nielsen, 1992]. Two regimes of particle transport at near-critical conditions are indicated. For very fine
sand (62 – 125 mm), values of the Shields parameter required
for suspension, qs, are less than the values required for
initiation of motion, qcr. As a result, very fine sand enters
directly into continuous suspension transport upon mobilization, and equations (2) – (4) can be expected to provide a
reasonable prediction of the reference concentration.
[11] In contrast, qs is greater than qcr for sediment with
a characteristic diameter in excess of 100– 200 mm, and
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S = u*/ws; and (3) the Shields parameter introduced in
equation (5).
[13] Only two parameters are required when considering
initiation of sediment because one is not an independent
variable, but a function of other two parameters. Yalin
[1963] considered the equality between the grain weight,
G ¼ af ðrs rÞgds3 ;
ð8aÞ
and the flow resistance, R, of a uniformly falling grain
R ¼ f ðws ds =nÞrds2 w2s ;
ð8bÞ
where af is a constant. The equality gives
rw2s
f ws ds
¼ 1;
n
ðrs rÞgds
ð8cÞ
2
f Re ws qsf ws ¼ 1;
u*
u*
ð8dÞ
that is,
where f = f/af. From equation (8d) it is evident that
u* =ws ¼ f Re; qsf :
ð8eÞ
Therefore the inverse Rouse number is conventionally
ignored [Liu, 1958; Yalin, 1963]. We suggest instead that in
the light of the perspectives developed above, it is more
sensible to neglect the grain Reynolds number but include
the inverse Rouse number:
Cr ¼ j u* =ws ; qsf :
Figure 1. (a) Values of the Shields parameter required for
the initiation of motion and initiation of suspension as a
function of quartz grain diameter. (b) Corresponding values
of critical shear velocity and settling velocity required for
initiation of motion and suspension, respectively. See color
version of this figure in the HTML.
thus material in this larger size range is typically transported as bed load by means of hops, rolls and saltation
over the bed upon mobilization. The mobilized sediment
enters into suspension only when the fluid force exceeds
the settling velocity of the material. We propose that, in
natural sediments comprising a range of sizes spanning
100 mm, only a fraction of the total amount mobilized is
actually available for suspended sediment transport and
thus should be included in the reference concentration Cr.
[12] A formal statement of our ideas is as follows.
The amount of sediment available for suspension
depends upon the forces of fluid flow and the resistance
of the particle to that flow. Previous analyses have
identified a suite of key, dimensionless parameters that
should be considered [Liu, 1958; Collins and Rigler,
1982]. These parameters are (1) the grain Reynolds
number Re = u*ds/n; (2) the inverse Rouse number
ð9Þ
In the following section we evaluate j (u*/ws, qsf)
empirically.
3. Data and Environmental Conditions
[14] To examine appropriate reference concentrations applicable over wide-ranging dynamic conditions and sand
sizes, we employed three data sets. Two data sets were
collected at 15 and 20 m depth on the inner shelf off
Dounreay, Scotland during 1997 and 2001, and one data
set was obtained at 13 m depth during 1996 on the inner shelf
off Duck, North Carolina, USA. Data acquired at 15 and 20 m
depth off Dounreay are referred to as DY97 and DY01,
respectively, while data obtained from Duck are referred to as
DK96 hereafter. Table 1 summarizes environmental and
experimental characteristics for the three experiments. The
Duck site is strongly influenced by both waves and winddriven currents, while the Dounreay site is characterized as a
wave-dominated environment. The median grain size of bed
sediment is 120 mm at Duck, and is 350 and 290 mm at 15 and
20 m depth at Dounreay, respectively. The Duck data are
described in more detail by Lee et al. [2002, 2003].
[15] Figure 2 shows time series of environmental conditions for the three experiments including mean current
velocity (uc), near-bed orbital velocity (ub), wave period
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Table 1. Characteristics of the Three Data Sets
Data
Site
DY01
DY97
DK96
Dounreay,
Dounreay,
Duck, North
Scotland, UK Scotland, UK Carolina, USA
2001
1997
1996
20
15
13
290
350
120
Year
Depth, m
Median size of bed
sediment, mm
3.7
ws, cm s1
1.38
u cr , cm s1
*
Flow sensor
Type
Nortek Vector
Height, cm
30
Sample rate, Hz
5
Sample duration, min
7
Sampling interval, hours
1.5
ABS
Acoustic frequency, MHz
2
Height, cm
80
Sample rate, Hz
5
Sample duration, min
7
Sampling interval, hours
1.5
16.7
Mean ub, cm s1
9.5
Mean uc, cm s1
4.8
1.46
1.0
1.22
EMCM
40
5
9
1.5
EMCM
98
1
12
2
2
82
2.5
4
1.5
18.9
9.5
2
88
5
12
2
19.0
14.0
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sediment trap fixed to one of the legs of the deployment
frames or using sand taken from the bottom by divers at the
beginning of the experiment. During calibration, the backscatter signals 54 cm below the ABS transducers were
inverted to obtain suspended sediment concentration.
Figure 3 compares ABS measurements to suction samples
collected during the laboratory calibrations.
[19] In order to determine reference concentration from
observed concentration profiles, the bed level must be
identified. The distance to the seabed during the field
experiment rarely corresponds to the nominal seabed range
chosen during the instrumentation setup because of passage
of bed forms and settling of the frame during the experiment. We used the method of Green and Black [1999] to
determine the seabed position for each burst. The burstaveraged concentration profiles usually exhibit an apparent,
strong acoustic signal extending below the real bed position
due to the ABS response to the very strong sound reflection
at the surface of the bed. Above the strongest return there
exists a break-in slope on a semilog plot of range from the
ABS against concentration (Figure 4). The break-in slope
(T), and shear velocities (u*sf, u*cw and u*c). Estimation of
shear velocities via the wave-current interaction model of
Grant and Madsen [1986] follow the methods described by
Lee et al. [2002]. The skin-friction components, u*sf, is the
shear velocity responsible for mobilization of sediment
particles at the bed; u*cw is the amplitude of shear velocity
due to the combined effect of waves and currents inside the
wave boundary layer, and u*c is wave-averaged shear velocity just above the wave boundary layer.
[16] The environmental conditions during Dounreay 1997
and 2001 are similar in that currents were driven mainly by
tides. Current speed approached 25 cm/s during spring tides,
but intervals of relatively high wave energy and significant
sediment transport occurred during neap tides. Thus the
Dounreay site is characterized as wave dominated. In
contrast, the Duck site was subject to an extratropical storm
during the experiment. Mean currents were dominated by
the wind-driven component, and the maximum reached up
to 50 cm/s. During all three experiments considered here,
the maximum near-bed wave orbital velocity exceeded
60 cm/s. The current contribution to sediment suspension
was very weak at Dounreay, but played a significant role in
suspending sediment during the storm event at Duck [Lee et
al., 2002].
4. Determination of Reference Concentration
From ABS Measurements
[17] The ABSs were mounted at a nominal elevation
above the bed looking downward (Table 1). Range-gating
the backscattered acoustic signal allowed the sediment
concentration profile to be estimated at 103, 81 and 109
range bins for DY01, DY97 and DK96, respectively, with a
vertical resolution of 1 cm. The sampling rate, sample
duration and sample intervals are tabulated in Table 1. A
detailed description and theory of the ABS technique can be
found in the work of Thorne et al. [1993].
[18] The ABSs were calibrated in a laboratory tank at the
University of East Anglia using sand collected in a passive
Figure 2. Time series of the environmental conditions
during the three experiments considered here (Dounreay
2001, Dounreay 1997, and Duck 1996). (a, d, and g) The
thick line shows near-bed wave orbital velocity, ub; the thin
line shows wave period, T. (b, e, and h) The thick line shows
the along-shore component of current velocity; the thin line
shows the cross-shore component of current velocity. (c, f,
and i) The thick line shows u*sf; the thin solid line shows
u*c; and the thin dashed line shows u*cw. See color version
of this figure in the HTML.
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Figure 3. Comparison of concentrations measured by
suction and ABS. DY01, DY97, and DK96 represent data
for Dounreay 2001, Dounreay 1997, and Duck 1996,
respectively. See color version of this figure in the HTML.
point is taken as the lowest echo uncontaminated by
backscatter from the seabed. The vertical and horizontal
arrows in Figure 4 indicate bed level and a measurement
point 1 (±0.5) cm above the bed (ab hereafter), respectively.
The concentration profiles in Figure 4 are from the first
burst after the initial frame deployment in each experiment.
[20] The ABS data include periods of no or little suspension. In order to ensure sufficiently strong energy conditions
to suspend sediment, we screened the ABS data based on
how well the suspension in the lowest 20 cm fit a onelayered Rouse-type equation. The regression model used
here is expressed as
C ¼ Cr ð z=zr ÞP ;
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reference concentration equals the static bed concentration
at a level slightly above the bed, either some small multiple
of the grain size or at the height of the bed roughness length
[Van Rijn, 1984; Dyer, 1986]. The reference height is set at
1 cm ab in this study for general consistency among sites.
Perhaps most importantly, an approximate elevation of 1 cm
simply represents the lowermost extent of direct observations. Additional analyses using reference concentrations
evaluated by the regression model (10) at different reference
heights indicate that the precise value chosen for zr does not
affect the overall substance of our results in the following
sections (see section 6).
[ 22 ] Figure 5 shows the fraction of variance (R 2 )
accounted for by regression of a one-layered Rouse equation to backscatter between 1 and 20 cm ab as a function of
the relative intensity of the sediment-transporting flow for
each of the 1643 profiles. In Figure 5, as in Figures 6 –10,
the circles, crosses and squares represent values pertaining
to Dounreay 2001 (DY01), Dounreay 1997 (DY97) and
Duck 1996 (DK96), respectively. Figure 5 reveals that, in
general, vertical profiles of suspended sediment are well
described by equation (10) with relative performance increasing with the overall intensity of a flow. In the following
analysis we consider the 706 cases for which R2 0.95. A
representative subsample of reference concentration is tabulated in Table 2 along with wave orbital velocity, wave
period, current velocity, u*sf, u*c and u*cw.
5. Examination of Existing and New Approach
for Reference Concentration
[23] In this section we examine the existing formulae for
reference concentration using the inferred reference con-
ð10Þ
where P is the Rouse parameter (= ws/ku*) and k 0.4 is
von Karman’s constant. The characteristic shear velocity,
u*, can be either u*c or u*cw depending on dynamical flow
conditions [Lee et al., 2002, 2003]. The use of a one-layered
Rouse-type equation in regression analysis of near-bed ABS
concentration is consistent with previous observations that
has shown the inferred eddy diffusivity profile to increase
linearly close to the bed [Vincent and Downing, 1994;
Sheng and Hay, 1995; Vincent and Osborne, 1995; Lee et
al., 2002, 2003], and that separate slopes to the concentration profile associated with u*c and u*cw above and within
the wave boundary layer cannot be resolved using ABS.
[21] In the concentration profiles it is necessary to specify
a reference height at which to define the reference concentration, Cr. Because the concentration increases rapidly
toward the bed, the reference height must be defined very
close to the bed. If the reference height is taken at zr = 0,
however, the diffusion equation predicts infinite concentration. Therefore it has been commonly assumed that the
Figure 4. Representative burst-averaged ABS profiles of
suspended sediment from each of the DY01, DY97, and
DK96 deployments, demonstrating the criterion for determining bed level. Vertical and horizontal arrows indicate
bed level and 1 cm ab, respectively. DY01, Dounreay 2001;
DY97, Dounreay 1997; DK96, Duck 1996. See color
version of this figure in the HTML.
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Figure 5. Fraction of variance (R2) explained by a
regression model of burst-averaged sediment concentration
in g/L as a function of u*sf /ws. The dotted line indicates the
criteria for data inclusion (R2 > 0.95). DY01, Dounreay
2001; DY97, Dounreay 1997; DK96, Duck 1996. See color
version of this figure in the HTML.
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Figure 7. Resuspension coefficient as a function of the
skin-friction Shields parameter. DY01, Dounreay 2001;
DY97, Dounreay 1997; DK96, Duck 1996. See color
version of this figure in the HTML.
centrations in the previous section. Figure 6 shows inferred
values of the reference concentration Cr at zr = 1 cm ab as a
function of excess shear stress. In general, equation (2)
captures a proportional relationship between Cr and excess
shear stress, but fails to accommodate observations from
different environments satisfactorily. The empirical formulae
(3) introduced by Van Rijn [1984] results in a pattern similar
to Figure 6. The discrepancy among different data sets is
concealed in the resuspension coefficient. Indeed, as shown
in Figure 7, inferred values of the resuspension coefficient go
(determined by forcing a best-fit to equation (2)) span the
range 105 to 102 and, at Dounreay at least, appear to
decrease with increasing flow intensity. In order to examine the sensitivity of these results to the chosen reference
height, Cr was also estimated by extrapolating observed
concentration to zr = 2.5ds. As shown in Figure 8, defining
zr = 2.5ds causes the scatter associated with a plot of Cr
versus excess shear stress to increase, especially for DK96.
Since the discrepancy among different data sets is even
worse with data estimated at 2.5ds, the selection of a fixed
elevation above the bed is probably not responsible for
the discrepancy among data sets obtained from different
environments.
Figure 6. Reference concentration in g/L as a function of
excess shear stress. DY01, Dounreay 2001; DY97,
Dounreay 1997; DK96, Duck 1996. See color version of
this figure in the HTML.
Figure 8. Reference concentration in g/L estimated at
2.5ds by the regression model of equation (10) as a function
of excess shear stress. DY01, Dounreay 2001; DY97,
Dounreay 1997; DK96, Duck 1996. See color version of
this figure in the HTML.
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regression analysis. Table 3 tabulates best-fit coefficients,
95% confidence intervals, and the fraction of variance (R2)
accounted for by each regression. The case with the highest
R2 value is model V (R2 = 0.89), simply because it has the
largest number of best-fit parameters. Since model V is
physically unrealistic and also the most complex, we reject
this model. Four models have same R2 value of 0.87, which
include models II, VI, VII, and VIII. Because the relative
performance of these models cannot be distinguished statistically, we chose model II, which is the simplest and is
consistent with the physical reasoning behind equation (11).
6. Discussion and Conclusions
Figure 9. Reference concentration in g/L as a function of
the Shields parameter raised to the power of 3. DY01,
Dounreay 2001; DY97, Dounreay 1997; DK96, Duck 1996.
See color version of this figure in the HTML.
[24] Figure 9 shows inferred values of the reference
concentration as a function of the skin-friction Shields
parameter raised to the power of 3. Nielsen’s model also
captures a proportional relationship between Cr and flow
intensity, but fails to capture the differences among data sets
to a satisfactory degree.
[25] Figure 10 displays the reference concentration in g/L
as a function of the product of the Shields parameter and the
inverse Rouse number. Unlike Figures 6 and 9, the dimensionless parameter accommodates all observations from
different environments. The empirical relationship for the
reference concentration, indicated by the solid line in
Figure 10, was found by fitting
B
u
Cr ¼ A qsf *sf ;
ws
[27] The important result of this study is the introduction
of a more universal, empirical equation for reference concentration for wide-ranging hydrodynamic and sedimentary
environments. Although the existing formulae for reference
concentration exhibited a proportional relationship (Figures
6 and 9) between shear stress and Cr, these formulae could
not reconcile the observations obtained from different
environments. The primary reason these existing formulae
fail to accommodate all the observations is their sole
dependence on threshold stress for initiation of motion of
particles when a threshold condition for suspension is also
required. Theoretical arguments suggest that the reference
concentration is dependent on the inverse Rouse number
and the grain Reynolds number, as well as the threshold
stress. However, these three parameters are dependent each
other and only two parameters are required when considering initiation of sediment motion and suspension. In fact,
the relative performance of regression models using these
parameters was statistically indistinguishable (Table 3). We
recommend using the reference concentration that is a
function of the product of the Shields parameter and inverse
ð11aÞ
where Cr in g/L is evaluated at 1 cm ab and, from regression
analysis,
A ¼ 2:58 1:17
ð11bÞ
B ¼ 1:45 0:04;
ð11cÞ
and
where the ± values indicate the 95% confidence interval of
the estimate. The regression model of equation (11)
accounts for 87% of the observed variability in Cr. The
success of equation (11) in accommodating a wide range of
dynamic conditions and sediment properties is consistent
with our initial proposal that Cr under combined waves and
currents is related to relative bed shear stress (represented
by qsf) adjusted for the relative intensity of sediment
suspension (represented by u*sf /ws).
[26] For comparison, various combinations of three
parameters considered in this study were subjected to
Figure 10. Reference concentration in g/L as a function of
the product of the skin-friction Shields parameter and
inverse Rouse number (u*sf /ws). The solid line indicates the
regression of the form given by equation (11) and specified
in the figure. DY01, Dounreay 2001; DY97, Dounreay
1997; DK96, Duck 1996. See color version of this figure in
the HTML.
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Table 2. Representative Samples of the Observed Reference
Concentration
uc
ub
T
u sf
*
20.34
18.94
22.91
22.48
14.23
18.05
29.19
24.93
22.85
17.73
24.07
25.04
20.95
30.84
19.73
23.75
16.88
40.62
40.73
52.35
39.38
41.50
29.70
27.11
20.47
31.10
52.46
64.03
57.41
31.39
35.10
22.35
20.24
20.42
22.65
35.55
35.92
34.04
21.67
37.16
36.54
40.24
29.22
31.69
27.02
21.94
28.15
20.39
17.59
16.50
18.23
16.42
10.86
23.11
8.65
9.51
28.01
28.18
5.15
16.97
11.50
8.60
11.34
12.95
12.02
21.34
11.63
14.45
19.27
1.95
14.37
6.17
9.71
8.61
6.48
2.26
7.70
9.96
6.15
13.09
6.82
8.38
13.71
5.41
7.54
14.28
17.16
10.63
9.76
7.60
11.06
8.95
7.38
11.28
10.60
10.99
16.03
3.46
11.36
2.22
12.72
20.80
20.90
12.53
9.79
9.85
9.91
9.99
8.46
10.87
9.29
10.14
8.64
7.86
10.85
9.83
9.17
10.54
9.38
10.15
9.67
11.08
9.43
9.97
9.10
10.48
10.06
9.75
8.41
8.28
10.79
11.52
9.80
9.41
9.46
9.29
12.05
10.92
11.36
13.36
12.22
13.20
12.15
12.96
12.08
11.49
10.91
11.19
10.53
10.11
9.63
10.27
9.77
9.12
10.84
9.11
1.54
1.68
1.67
1.66
1.48
1.60
2.12
1.94
1.72
1.46
1.78
1.86
1.70
2.43
1.60
1.89
1.45
2.70
2.83
3.40
2.74
2.83
2.14
1.93
1.63
2.24
3.44
4.07
3.77
2.25
2.49
1.68
1.47
1.55
1.69
2.38
2.43
2.29
1.56
2.47
2.43
2.68
2.05
2.17
1.97
1.61
2.06
1.52
1.41
1.55
1.59
1.39
u cw u c
*
*
DY01
3.66 1.34
3.85 2.29
3.77 1.15
3.75 1.22
3.51 2.50
3.57 2.53
4.19 0.86
3.96 1.83
4.13 1.44
3.77 1.14
3.60 1.34
3.94 1.52
3.99 1.50
4.02 2.05
3.82 1.44
3.96 1.67
3.51 1.96
4.18 0.49
4.62 1.65
5.21 1.03
4.58 1.28
4.43 1.19
3.85 0.95
3.93 0.49
4.04 1.11
4.66 1.34
5.14 1.09
5.92 1.80
5.77 1.22
4.11 1.13
4.22 1.54
3.94 0.86
3.15 0.98
3.46 1.57
3.43 1.74
3.54 1.22
3.72 1.18
3.46 0.98
3.15 1.25
3.67 1.11
3.73 0.99
4.09 1.34
3.48 1.21
3.58 1.25
3.70 1.67
3.68 0.63
4.07 1.39
3.56 0.48
3.46 1.47
3.70 2.11
3.58 2.08
3.47 1.45
22.46
22.08
42.21
29.47
18.04
18.93
23.92
22.29
19.60
26.99
30.83
56.17
27.06
23.56
20.92
16.49
8.70
12.90
3.90
10.12
15.02
8.02
1.93
11.96
5.47
1.41
7.15
11.59
16.76
3.24
9.88
8.41
10.56
9.84
8.49
8.94
8.72
12.77
11.65
8.47
7.92
9.42
7.60
7.20
8.02
1.88
1.77
2.94
2.18
1.52
1.67
1.86
1.65
1.57
2.06
2.33
3.80
2.14
1.99
1.68
DY97
4.72 1.95
4.74 1.21
5.31 1.63
5.25 0.77
4.07 1.26
4.31 1.73
4.93 1.17
4.00 0.45
3.98 1.47
5.37 0.94
5.99 0.44
6.70 1.24
5.65 1.59
5.26 1.97
4.57 0.60
Cr at 1 cm ab
Cr at 2.5ds
0.008
0.006
0.014
0.018
0.004
0.007
0.046
0.016
0.027
0.004
0.032
0.018
0.014
0.122
0.015
0.023
0.004
0.049
0.024
0.088
0.043
0.067
0.021
0.040
0.038
0.150
0.119
0.386
0.060
0.066
0.058
0.012
0.015
0.012
0.015
0.040
0.055
0.130
0.016
0.050
0.028
0.086
0.033
0.026
0.030
0.020
0.060
0.021
0.006
0.007
0.008
0.004
0.133
0.034
0.265
0.403
0.043
0.051
1.309
0.123
0.802
0.039
0.406
0.213
0.117
1.879
0.135
0.199
0.028
0.322
0.202
0.819
0.514
1.372
0.351
0.450
0.519
4.657
0.794
8.491
0.203
1.158
2.046
0.132
0.257
0.133
0.145
1.423
0.828
6.570
0.154
0.672
0.185
0.428
0.381
0.205
0.247
0.362
1.087
0.412
0.051
0.064
0.073
0.038
0.037
0.009
0.319
0.081
0.007
0.035
0.036
0.016
0.016
0.043
0.154
0.254
0.101
0.131
0.013
0.314
0.055
2.031
0.420
0.039
0.413
0.158
0.084
0.073
0.470
0.808
1.810
0.612
1.667
0.067
C02021
Table 2. (continued)
uc
ub
T
u sf
*
15.19
20.13
18.26
18.79
19.14
18.84
25.47
36.89
43.19
40.31
37.89
38.50
44.24
41.39
32.28
41.38
35.14
33.16
35.62
32.38
63.36
35.41
27.52
16.34
19.33
31.50
25.48
28.08
27.35
23.11
9.20
4.96
1.06
5.03
5.45
36.35
49.01
41.70
44.36
43.83
15.75
31.67
40.44
7.07
34.75
23.14
13.08
13.92
16.39
10.05
16.56
26.10
23.45
17.60
6.03
5.31
3.28
7.01
8.07
7.45
8.11
8.17
9.40
9.23
8.15
7.42
7.64
8.46
8.67
7.96
8.47
7.81
8.09
8.23
8.78
8.85
7.76
8.02
10.05
10.38
9.40
7.64
11.08
10.94
10.86
11.49
12.10
1.35
1.48
1.32
1.30
1.31
1.30
2.02
2.86
3.17
2.74
2.61
2.53
2.83
2.77
2.14
2.73
2.43
2.13
2.37
2.15
3.70
2.29
1.95
1.35
1.37
2.00
1.66
1.78
1.77
u cw u c
*
*
DK96
2.90 1.64
3.18 0.84
2.88 0.52
2.84 0.16
2.55 0.50
2.59 0.53
3.30 2.18
4.51 2.96
4.94 2.79
4.14 2.68
3.94 2.61
3.96 1.30
4.29 2.17
4.28 2.56
3.43 0.70
4.17 2.28
3.71 1.68
3.30 1.06
3.77 1.17
3.45 1.26
5.69 1.14
3.32 1.27
3.03 1.68
2.98 1.66
2.22 1.16
2.91 0.60
2.54 0.51
2.62 0.37
2.55 0.64
Cr at 1 cm ab
Cr at 2.5ds
0.148
0.448
0.075
0.169
0.084
0.115
0.179
1.119
0.697
2.788
2.191
2.140
4.889
1.629
0.832
5.718
0.665
5.561
7.997
0.601
8.827
0.954
0.387
0.074
0.131
2.910
0.145
0.370
0.223
12.678
92.404
0.761
33.911
2.440
3.442
15.064
16.569
6.745
187.536
103.738
97.112
378.422
48.148
5.986
647.650
16.627
428.270
1217.576
11.160
272.798
31.182
11.226
0.338
2.046
73.429
1.559
3.752
4.332
Rouse number because it is the simplest and consistent with
our physical reasoning. In this respect, the inverse Rouse
number modifies the total amount of mobile sediment
(related to the Shields parameter) to account for that fraction
available for suspension. As a result, the new formula
accommodates all the observations from the different
environments.
[28] It is important to emphasize that equation (11)
pertains to a reference height of zr = 1 cm. Since there is
no general consensus on the most appropriate reference
height, we set the reference height at the lowest level to
which suspended sediment concentration could be estimated
with confidence through direct observation. Other elevations could be chosen with theoretical justification. To
examine the general consistency of the approach developed
here, we considered different reference heights in a broader
analysis. For example, concentrations at 2.5ds were estimated
by the regression model of equation (10) and were plotted
as a function of the product of Shields parameter and
inverse Rouse number (Figure 11). In contrast with the
Table 3. Best-Fit Coefficients, Confidence Interval, and Fraction
of Variance (R2) Accounted for by Each Regression
95% Confidence
Interval
Coefficient
Model
A
B
I
A(qsf SRe)
II
A(qsf S)B
III
A(qsf Re)B
IV
A(SRe)B
V
AqsfB SCReD
VI
Aq sfB SC
VII AqsfB ReC
VIII AS BReC
8 of 10
0.39
2.58
0.21
0.009
1037
1.51
91
0.05
B
C
D
A
1.22
1.18
1.45
1.17
1.34
1.28
2.37
1.12
29 39 24 9 105
1.19 1.77
1.39
2.65 1.12
1.32
3.22 0.95
1.13
B
0.05
0.04
0.11
0.12
4.86
0.14
0.08
0.09
C
D
R2
0.74
0.87
0.45
0.65
5.89 3.81 0.89
0.18
0.87
0.12
0.87
0.11
0.87
C02021
LEE ET AL.: REFERENCE CONCENTRATION ON THE INNER SHELF
C02021
influence of bed forms on the magnitude of reference
concentration is clear [Green and Black, 1999; Webb and
Vincent, 1999]. Therefore we acknowledge that accommodation of bed form geometry will contribute to the improved
performance of predictive expressions of the form given by
equation (11). Such improvements are a focus of ongoing
study.
[30] Acknowledgments. G.L. and Dounreay deployments were supported by the (UK) Atomic Energy Authority. W.B.D. was supported by the
(UK) Natural Environmental Research Council. Participation by C.T.F. and
field studies at Duck, NC, were supported by the (US) National Science
Foundation.
References
Figure 11. Reference concentration in g/L estimated at
2.5ds by the regression model of equation (10) as a function
of the product of the skin-friction Shields parameter and
inverse Rouse number. DY01, Dounreay 2001; DY97,
Dounreay 1997; DK96, Duck 1996. See color version of
this figure in the HTML.
relationship shown in Figure 10, the scatter of concentration data, especially for DK96, increased (R2 = 0.78).
Nonetheless, the scaling parameter proposed here accommodates all estimates from wide-ranging environments.
Thus the essential outcome of our analysis is robust, and
independent of the exact reference height chosen. We
acknowledge, of course, that improved expressions similar
to equation (11) are likely to come with improved theoretical constraints on the appropriate reference height and
technical ability to measure sediment concentration at that
height.
[29] Recent studies suggested that the influence of bed
forms must be taken into account in order to improve the
model prediction on reference concentration [Webb and
Vincent, 1999]. The significance of bed form influence on
estimates of reference concentration was clearly demonstrated by Green and Black [1999]. In that study, they
observed two groups of reference concentration that were
separated by bed form types. The two bed types included
large hummocks formed during energetic wave conditions
and rippled bed formed during less energetic conditions.
They applied a correction for flow contractions over ripples
by multiplying the Shields parameter by (1 ph/l)2, where
h and l are observed ripple height and ripple length. This
enabled the model based on the skin-friction Shields
parameter to adequately predict the reference concentration
over a range of bed forms. In this study, we neglected the
effect of bed forms on the magnitude of reference concentration because our data could not adequately quantify bed
form variability. First, we did not observe distinct groups of
reference concentration (compare Figures 6, 9, and 10) that
suggested the influence of bed forms as clearly as concentration data shown by Green and Black [1999, Figure 5].
Second, there were no direct observations of bed forms
collected during our experiments. Nonetheless, the potential
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C02021
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W. B. Dade, Department of Earth Sciences, Dartmouth College, Fairchild
Hall HB 6105, Hanover, NH 03755, USA. ([email protected])
C. T. Friedrichs, Virginia Institute of Marine Science, College of William
and Mary, P. O. Box 1346, Route 1208, Greate Road, Gloucester Point, VA
23062, USA. ([email protected])
G. Lee, Korea Ocean Research and Development Institute, 1270 Sadong,
Ansan 425-744, Korea. ([email protected])
C. E. Vincent, School of Environmental Sciences, University of East
Anglia, Norwich NR4 6TJ, UK. ([email protected])
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